Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures

2005 ◽  
Author(s):  
Brian J. Olson ◽  
Steve W. Shaw ◽  
Christophe Pierre
Keyword(s):  
Equations Of Motion ◽  
Flexible Structures ◽  
Closed Form Solution ◽  
Nonlinear Effects ◽  
Form Solution ◽  
Vibration Absorbers ◽  
Bladed Disk ◽  
Tuned Vibration Absorbers ◽  
Engine Order ◽  
Bladed Disk Assembly

This paper investigates the use of order-tuned absorbers to attenuate vibrations of flexible blades in a bladed disk assembly subjected to engine order excitation. The blades are modeled by a cyclic chain of N oscillators, and a single vibration absorber is fitted to each blade. These absorbers exploit the centrifugal field arising from rotation so that they are tuned to a given order of rotation, rather than to a fixed frequency. A standard change of coordinates based on the cyclic symmetry of the system essentially decouples the governing equations of motion, yielding a closed form solution for the steady-state response of the overall system. These results show that optimal reduction of blade vibrations is achieved by tuning the absorbers to the excitation order n, but that the resulting system is highly sensitive to small perturbations. Intentional detuning (meaning that the absorbers are slightly over- or under-tuned relative to n) can be implemented to improve the robustness of the design. It is shown that by slightly undertuning the absorbers there are no system resonances near the excitation order of interest and that the resulting system is robust to mistuning (i.e., small random uncertainties in the system parameters) of the absorbers and/or blades. These results offer a basic understanding of the dynamics of a bladed disk assembly fitted with order-tuned vibration absorbers, and serve as a first step to the investigation of more realistic models, where, for example, imperfections and nonlinear effects are considered, and multi-DOF and general-path absorbers are employed.

Toward a Minimal Dynamic Model of a 6-DOF Parallel Robot

1997 ◽  
Author(s):  
L. Beji ◽  
M. Pascal ◽  
P. Joli
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Parallel Robot ◽  
Form Solution ◽  
Lagrangian Formalism ◽  
Inverse Kinematic Problem ◽  
Geometrical Properties

Abstract In this paper, an architecture of a six degrees of freedom (dof) parallel robot and three limbs is described. The robot is called Space Manipulator (SM). In a first step, the inverse kinematic problem for the robot is solved in closed form solution. Further, we need to inverse only a 3 × 3 passive jacobian matrix to solve the direct kinematic problem. In a second step, the dynamic equations are derived by using the Lagrangian formalism where the coordinates are the passive and active joint coordinates. Based on geometrical properties of the robot, the equations of motion are derived in terms of only 9 coordinates related by 3 kinematic constraints. The computational cost of the obtained dynamic model is reduced by using a minimum set of base inertial parameters.

Stability Analysis of Mechanical Seals With Two Flexibly Mounted Rotors

1998 ◽  
Vol 120 (2) ◽  
pp. 145-151 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Dynamic Properties ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fluid Film ◽  
Mechanical Seals ◽  
Stability Threshold ◽  
Seal Design ◽  
The Stability ◽  
Gyroscopic Effects

Dynamic stability is investigated for a mechanical seal configuration in which both seal elements are flexibly mounted to independently rotating shafts. The analysis is applicable to systems with both counterrotating and corotating shafts. The fluid film effects are modeled using rotor dynamic coefficients, and the equations of motion are presented including the dynamic properties of the flexible support. A closed-form solution for the stability criteria is presented for the simplifled case in which the support damping is neglected. A method is presented for obtaining the stability threshold of the general case, including support damping. This method allows instant determination of the stability threshold for a fully-defined seal design. A parametric study of an example seal is presented to illustrate the method and to examine the effects of various parameters in the seal design upon the stability threshold. The fluid film properties in the example seal are shown to affect stability much more than the support properties. Rotors having the form of short disks are shown to benefit from gyroscopic effects which give them a larger range of stable operating speeds than long rotors. For seals with one long rotor, counterrotating operation is shown to be superior because the increased fluid stiffness transfers restoring moments from the short rotor to the long.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia
Keyword(s):  
Dynamical Systems ◽  
Closed Form ◽  
Programming Problem ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Nonholonomic Constraints ◽  
Form Solution ◽  
Constrained Motion ◽  
Constraint Forces

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

Design of the Ball Screw Mechanism for Optimal Efficiency

1989 ◽  
Author(s):  
M.-C. Lin ◽  
S. A. Velinsky ◽  
B. Ravani
Keyword(s):  
Closed Form ◽  
Static Analysis ◽  
Load Capacity ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Ball Screw ◽  
Exact Theory ◽  
Screw Mechanism ◽  
Extensive Computation

Abstract This paper develops theories for evaluating the efficiency of the ball screw mechanism and additionally, for designing this mechanism. Initially, a quasi-static analysis, which is similar to that of the early work in this area, is employed to evaluate efficiency. Dynamic forces, which are neglected by the quasi-static analysis, will have an effect on efficiency. Thus, an exact theory based on the simultaneous solution of both the Newton-Euler equations of motion and the relevant kinematic equations is employed to determine mechanism efficiency, as well as the steady-state motion of all components within the ball screw. However, the development of design methods based on this exact theory is difficult due to the extensive computation necessary and thus, an approximate closed-form representation, that still accounts for the ball screw dynamics, is derived. The validity of this closed-form solution is proven and it is then used in developing an optimum design methodology for the ball screw mechanism based on efficiency. Additionally, the self-braking condition is examined, as are load capacity considerations.

The Effects of Gravity on the Response of Centrifugal Pendulum Vibration Absorbers

2021 ◽  
pp. 1-61
Author(s):  
Darryl Tchokogoue ◽  
Ming Mu ◽  
Brian F. Feeny ◽  
Bruce K. Geist ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Operating Conditions ◽  
Vibration Absorbers ◽  
Cyclic Symmetry ◽  
Passive Devices ◽  
Engine Order ◽  
Gravity Effects ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Automotive Powertrain ◽  
Linearized Model

Abstract This paper describes the effects of gravity on the response of systems of identical, cyclically arranged, centrifugal pendulum vibration absorbers (CPVAs). CPVAs are passive devices composed of movable masses suspended on a rotor, suspended such that they reduce torsional vibrations at a given engine order. These absorbers are becoming prevalent in automotive powertrain components in order to expand fuel-efficient engine operating conditions. Gravitational effects acting on the absorbers can be important for a horizontal rotor/CPVA system spinning at relatively low rotation speeds, for example, during engine idle conditions. The main goal of this investigation is to predict the response of a CPVA/rotor system in the presence of gravity. A linearized model which includes the effects of gravity and an order n torque acting on the rotor is analyzed by exploiting the cyclic symmetry of the system. The results show that the N absorbers respond in one or more groups, where the absorbers in each group respond with identical waveforms but shifted phases. The number of groups depends on the engine order n and the ratio Nn. It is shown that there are special resonant effects if the engine order is n = 1 or n = 2, the latter of which is particularly important in applications. In addition, it is shown that for N > 1 the rotor response is not affected by gravity, due to the symmetry of the gravity effects. The analytical predictions are verified by direct simulations of the equations of motion.

Analysis and Synthesis of Dynamic Response of a Flexibly Coupled Rotor Assembly

1997 ◽  
Author(s):  
Valdas Chaika
Keyword(s):  
Equations Of Motion ◽  
Excitation Frequency ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Response ◽  
Wide Range ◽  
Analysis And Synthesis ◽  
Coupling Parameters ◽  
Elastic Couplings ◽  
Rotor Assembly

Abstract Torsional vibration of two flexibly coupled reciprocating machines is investigated. The rotors of the machines are connected by elastic couplings of several types. The system is excited by a harmonic torque. The excitation frequency is proportional to the rotational speed which varies within a wide range. The motion of the system is described by nonlinear ordinary differential equations. These are linearized for the specific case of the rotor assembly design. Applying impedance functions, a closed-form solution of the equations of motion is derived. Three different cases of the system response are analyzed in the frequency domain. The passive vibration control of the rotor assembly using the centrifugal coupling is investigated. An analytical synthesis technique of the coupling parameters is devised.

Seismic Response of Unanchored Fluid-Filled Tanks Using Finite Elements

1992 ◽  
Vol 114 (1) ◽  
pp. 74-79 ◽  
Author(s):  
Wei Yi ◽  
S. Natsiavas
Keyword(s):  
Finite Elements ◽  
Seismic Response ◽  
Analytical Procedure ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Element Model ◽  
Form Solution ◽  
Structural Domain ◽  
Hydrodynamic Response ◽  
Strong Ground

A finite element model is presented for the seismic response of liquid-filled tanks. This type of analysis is complicated for unanchored tanks, because the bases of these tanks separate from their foundations during strong ground motion. This changes the dynamic behavior of these structures considerably and may result in severe loading. The analysis starts by geometrically discretizing the shell structure using cylindrical finite elements. Then, application of Hamilton’s principle in the structural domain yields the equations of motion for the coupled fluid/structure system. The foregoing analytical procedure employs the closed-form solution for the hydrodynamic response problem, resulting in a compact system of equations of motion. Primary attention is paid to the formulation of the nonlinear base uplift problem. Effects due to shell and ground flexibility are also included.

The Sliding With Coulomb Friction of a Rigid Indentor Over a Power-Law Inhomogeneous Linearly Viscoelastic Half-Plane

1984 ◽  
Vol 51 (2) ◽  
pp. 289-293 ◽  
Author(s):  
J. R. Walton
Keyword(s):  
Shear Modulus ◽  
Half Plane ◽  
Power Law ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Plane Boundary ◽  
Physical Parameters ◽  
Title Problem ◽  
Rigid Indentor

In a previous paper, the title problem was solved for a homogeneous power-law linearly viscoelastic half-plane. Such material has a constant Poisson’s ratio and a shear modulus with a power-law dependence on time. In this paper, the shear modulus is assumed also to have a power-law dependence on depth from the half-plane boundary. As in the earlier paper, only a quasi-static analysis is presented, that is, the enertial terms in the equations of motion are not retained and the indentor is assumed to slide with constant speed. The resulting boundary value problem is reduced to a generalized Abel integral equation. A simple closed-form solution is obtained from which all relevant physical parameters are easily computed.

Effects of the Nature of Excitation on the Response of a Mistuned Bladed Disk Assembly

2002 ◽  
Author(s):  
D. Cha ◽  
A. Sinha
Keyword(s):  
Narrow Band ◽  
Analytical Techniques ◽  
Bladed Disk ◽  
Engine Order ◽  
Bladed Disk Assembly ◽  
Time Invariant

In this paper, responses of a mistuned bladed disk assembly are examined and compared for three types of excitations: uncorrelated narrow band random excitations, correlated narrow band random excitations and sinusoidal excitations with unknown (time-invariant and random) amplitudes. Analytical techniques are also developed to compute the statistics of responses for these types of excitations. Effects of correlations of narrow band excitations are investigated in details. It has been found that the response statistics for correlated narrow band random excitations can be viewed in terms of the concepts related to the response to a deterministic engine order excitation.

Sign in / Sign up

Export Citation Format

Share Document